CONFIGURACIÓN SITIO A SITIO SEGÚN LABORATORIO CCNA

For a fluid of dynamic viscosity, p and kinematic viscosity, u, the mean shear stress, T, is defined by the energy dissipation rate, e, in the vessel. Thus, for instantaneous velocities acting parallel to the surface of an aggregate the local shear stress can be wriUen as

T: = H ( - ) \

assuming that the fluctuating velocities are separated by a distance equal to the aggregate diameter.

The rate of disruption of aggregates in a turbulent flow field is assumed to be equal to the product of the number concentration of the aggregates, the frequency of disruption of the aggregates by the instantaneous turbulent stresses and the number of primary particles eroded per disruption (Parker et aL, 1972; Brown and Glatz, 1987; Ayazi Shamlou et a l , 1990).

Assuming the protein aggregates to have a mean mechanical strength, a , the rate of breakage following collisions between the aggregates and the turbulent eddies is determined by the fraction of the eddies having an energy in excess o f that holding the primary particles together. The rate of breakage of aggregates having an initial diameter, d f , can be expressed by

B(df) = f ( d f ) U f (3.23)

where nf is the number concentration of the aggregates, f(df) is the eddy- aggregate collision frequency and ^ is the probability of a collision leading to breakage. An expression can be obtained for f(df) by assuming that the frequency of eddy-aggregate interactions is equal to the frequency of occurrence of the smallest eddies, given by (Levich, 1962):

1 / E \ ^ ^

^(^f) “ 151/2 (3.24)

The probability of breakage following a collision between an eddy and an aggregate depends on both the mechanical strength of the aggregate and on the magnitude of the turbulent stresses acting on its surface. For a fixed intensity of agitation, this probability is expected to approach unity for weak aggregates and zero for strong ones. A general expression for this kind of variation is the exponential distribution law of probability,

% = exp ( - ^ ) (3.25)

Substituting this probabihty function in Equation 3.23 and replacing f(df) using Equation 3.24 gives the following expression for the rate of aggregate breakage.

1 f-<3\

B(df) = ^ l - j exp ) nf (3.26)

Equation 3.26 suggests that the rate of disruption of aggregates having sizes in the inertial subrange is expected to have a . order dependency with respect to the energy dissipation rate per unit mass and a first order dependency with respect to the volume concentration of the aggregates, nf.

External forces originating from the mechanical agitation of the liquid aie primarily responsible for two particles approaching one another. A successful collision requires the squeezing of the continuous liquid film out of the gap between the particles. As the gap reduces, the interparticle repulsive double­ layer forces and van der Waals attractive forces begin to operate on the particles, as portrayed earlier in Figure 2.7.

In Equation 3.26, a , is the mechanical strength of a single aggregate. The overall strength of an aggregate is given by the sum of the strength of the individual particles. An equation that provides an estimate of this strength is given by (Rumpf, 1962):

Aggregation occurs due to the attractive van der Waals forces between the particles.

In Equation 3.27, it is assumed that the only inter-particle forces between two primary particles within the aggregate are the van der Waals forces of attraction, F. For a pair of equal size primary particles of diameter, dp , and

with a separation distance of Hq , the van der Waals forces of attraction can

be expressed by:

A dp

where A is the Hamaker constant for the liquid-particle system. Additionally, in Equation 3.27, Iq. is the co-ordination number which, based on experimental observations (Kusters, 1991) appears to have a power-law dependency upon the volume fraction of solids within the aggregate, ([), given by

kp = X (t)Y (3.29)

with Xi=15 and Y=1.2. Substituting these values together with Equation 3.28 in Equation 3.27 gives:

a = 0 .4 4 ( |) 2 - 2 ^ ; f (3.30)

Hr*

Equation 3.30 offers a simple explanation of the mechanical strength of an aggregate. Its derivation assumes that the structure of aggregates is uniform with a constant porosity. Experimental data, however, suggest that most aggregates have a non-uniform structure and aggregate density decreases as aggregate size increases (Tambo and Watanabe, 1979).

Protein precipitates, as mentioned earlier, are characteristically open structures with high voidages. The properties of the aggregates result from the growth processes (Family and Landau, 1984) occurring in the reactor. Computer simulation of particle growth and theoretical work (Void, 1963)

suggest that protein precipitation results in the formation of scale-invariant fractal aggregates. Essentially, the structure of such an aggregate can be characterized by the magnitude of the fractal dimension, y. The number of primary particles, Np, within a fractal aggregate is related to its radius, Rf, by the following expression (Gregory, 1989):

Np = Rf''^

(3-31)

In practice the value of y may be obtained from the slope of a log-log plot of aggregate mass against its size as shown in Figure 3.7. The value of y varies between 1 for a linear aggregate and 3 for an aggregate with uniform density (Weitz and Huang, 1984). These computer simulation studies suggest that values of y are in the range of 1.6 to 2.0 for aggregation by cluster-cluster collisions, while aggregation via particle-particle collisions results in aggregates having values of y of about 2.5 (Ring, 1991). A fractal dimensionality of less than 3 indicates that the concentration of particles falls off with distance from the centre of the aggregate. Evidently, aggregation by cluster-cluster collisions produces aggregates which are less dense (i.e. greater porosity) on average than particle-particle aggregates. Additionally, the number, Np, of primary particles of radius, a, within a fractal aggregate is related to its radius, Rf, by the following equation (Sonntag and Russel,

1987):

(3.32)

Little experimental data exists on the fractal dimensionality of protein precipitates. However, in a recent publication (Appendix D3) regarding the analysis of the turbulent breakage process of protein aggregates studied in this

I

Aggregate size, df (m)

F ig u re 3.7. The slope o f a log-log plot o f aggregate mass versus aggregate size to give an indication o f the fractal dimension o f the aggregate, which in turn, provides important information on the internal structure o f the aggregate and on the likely mechanisms of the aggregation process.

work, it has been concluded that the protein precipitate results in the formation of scale-invariant fractal aggregates with a fractal dimensionality close to 2.2. This value indicates that protein growth during precipitation occurs by a combination of particle-particle and cluster-cluster collisions.

For Brownian aggregation of polystyrene spheres of 0.14 |im diameter in aqueous suspension the value of the constant k' in Equation 3.32 was found to be 0.455 and the value of the fractal dimensionality, y, was equal to 2.48. This is consistent with the experimental observations of the present study as well as previous findings (Bell et al., 1982) where the aggregate density has been shown to decrease with an increase in aggregate diameter.

Nevertheless, the strongest impact of the concept of fractal aggregates appears to be in studies on computer simulation of aggregate structure (Sutherland,

1967; Goodarznia, 1979; Family and Landau, 1984). More direct

experimental data on the fractal dimension of aggregates would be helpful in trying to understand its impact on aggregation and break-up processes. The variation of the volume fraction of solids with radial position, r, within a fractal aggregate has been shown to take the following form (Sonntag and Russel, 1986)

<f(r) = k' ( : ^ ) ( 3) ( ; ) ^ (3.33)

Equation 3.33 then provides a description of the distribution of the particles as a function of radial position within an aggregate and can be used together with Equation 3.30 to obtain an estimate of the distribution of the mechanical strength within the aggregate. However, since most experimental and theoretical evidence indicate that the breakage of protein aggregates occurs by

erosion of small fragments from the surface of the aggregates, (Bell and Dunnill, 1982a; Hoare 1982a & b; Glatz et al., 1986) it is therefore reasonable to suggest that it is the solids volume fraction close to the surface of the aggregate that determines whether, and if so to what extent, aggregate breakage occurs.

In the absence of data for the protein precipitates, setting y=2.48 and k'=0.455 in Equation 3.33, assuming protein primary particles to have a diameter of 0.5 mm and taking a typical 10 |xm protein aggregate gives a value for the surface volume fraction, $(Rf), of 0.038. Substituting this value into Equation 3.30, assuming a typical mean separation distance between primary particles, Hq, of 3 nm (Hogg, 1989) and a value of the Hamaker constant. A, of lO'^l J, gives a value of 0.26 N/m^ for the surface mechanical strength c(Rf) of the aggregate. For a mechanically stirred vessel of a standard configuration Lee and Brodkey (1987) suggest that fluid stress levels below about 0.75 N/m^ can be considered to be mild while stress levels above about 5 N/m^ are classed as high. Setting x=0.75 N/m^ and using the estimated value of the aggregate surface strength (a=0.26 N/m^), gives a value for the probability of breakage (ex p -a/t) of about 0.7 which suggests that even under relatively mild agitation, aggregate breakage is to be expected.

Assuming that the energy distribution in the vessel is uniform, the local energy dissipation rate can be replaced by its mean value, which in the case of turbulent flow is given by:

En, = KN3Di2 (3.34)

w h e r e k i s a p r o p o r t i o n a l i t y c o n s t a n t .

3/2 _ A FA/D \i2.2

_ _ /N"'" D i \ r -K A [(^ (R f)]- -,

B « l t ) « ( - : n 3 - ) “ P L H . : d , N » « D , p : ' 2 ; J ‘ '

Assuming that the aggregate size at the end of the ageing process, df, is significantly larger than the primary particle size, dp, and provided that breakage occurs via erosion of a single or small groups of primary particles from the surface of the aggregates, the breakage rate B(df) can be represented by the rate of change in the volume, Vf, of the aggregate. Thus

d df3 d df

B (df)o‘ - 5 r “ d f 2 - j;- (3.36)

Replacing B(df) in Equation 3.35 using Equation 3.36 and noting that for spherical aggregates with mean diameter, df, the number concentration is related to the volumetric concentration and the aggregate diameter, i.e. nf Cy/df^, and rearrangement gives

1 d df z N ^ ^ x r -KA[(|>(Rf)]^-^ 1 Cv _

df dt

\

)

Lhq2 dp N2/2 Df

pl/zj

The term on the left-hand-side of the equation which has units of reciprocal of time gives a measure of the frequency of breakage and can be obtained experimentally from the slope of aggregate diameter versus time. The right- hand-side of Equation 3.37 shows the influence on the breakage frequency of some of the most important material and operating parameters. It is these operating parameters which have been examined in this study.